We revisit the golfer's curse, which is the possibility that a golf ball can emerge from the cylindrical hole into which it has entered. Our analysis focuses on three constants of the motion. One of these is the energy, because we assume that the ball rolls without slipping on the inner wall of the hole, losing only a small amount of energy to rolling resistance; the other two are related to the angular momentum about the contact point of the ball with the inner wall of the hole. We develop an analysis of the motion of the ball and report measurements of the moment of inertia of a real golf ball. Solving the equation of motion along the vertical direction, we address the question of whether or not the ball could complete a vertical oscillation without reaching the bottom of the hole. We also present measurements of the dynamical friction for a golf ball and discuss dissipation in slip conditions. We conclude by proposing a challenge to golf players: to find a way to send a ball into a hole in order to make it emerge.
REFERENCES
1.
We could not find any online videos showing such an effect in a real golf game. The best we found is a video of a putt by Tiger Woods in the 2007 PGA Championship, where the ball entered the hole only slightly, made contact with its internal surface, and then emerged after an almost 180° turn (see <https://sports.yahoo.com/blogs/golf-devil-ball-golf/look-back-tiger-woods-putt-2007-pga-championship-213256799.html>, at time 1:10). Strictly speaking, this is not completely similar to the periodic vertical motion analysed here and in the references but is similar in that the ball seemed to fall before initiating a sudden upward motion and exiting the hole.
2.
J.
Wolstenholme
, Mathematical Problems on the First and Second Divisions on the Schedule of Subjects for the Cambridge Mathematical Tripos Examination
(
Macmillan
,
London
, 1878
), p. 461
.3.
E.
Routh
, A Treatise of the Dynamics of a System of Rigid Bodies
(
Macmillan
,
London
, 1905
), Vol.
II
, p. 165
.5.
J. I.
Neĭmark
and
N. A.
Fufaev
, “
Dynamics of nonholonomic systems
,” Transl. Math. Monogr., Am. Math. Soc.
33
, 95
(1972
).6.
7.
M.
Gualtieri
,
T.
Tokieda
,
L.
Advis-Gaete
,
B.
Carry
,
E.
Reffet
, and
C.
Guthmann
, “
Golfer's dilemma
,” Am. J. Phys.
74
, 497
–501
(2006
).8.
O.
Pujol
and
J.-Ph.
Pérez
, “
On a simple formulation of the golf ball paradox
,” Eur. J. Phys.
28
, 379
–384
(2007
).9.
J.-Ph.
Pérez
, Mécanique, Fondements et Applications
, 7th ed. (
Dunod
,
Paris
, 2014
), p. 278
(in French).10.
B. W.
Holmes
, “
Putting: How a golf ball and hole interact
,” Am. J. Phys.
59
, 129
–136
(1991
).11.
In the present work, it is assumed that
and Ωz are unrelated quantities. They might be related under two conditions: (1) if va were identified with the initial velocity
(viz., the initial velocity of C of the ball just after entering the hole), and (2) if
were taken to be 0. One, thus, would arrive at the conclusion that
; Eqs. (32) and (35) then would become
and
. It would follow that
and
. (We consider only the positive values of
) With the values of Table I, we have
and
, which would correspond to approaching velocities
m s−1 and
, in perfect agreement with Fig. 5. Because the interaction between the hole and the ball is complex (see Ref. 10), the conditions (1) and (2) are not obvious. It is, thus, reasonable and simpler from a pedagogical point of view to consider
and Ωz as independent.
12.
K.
Zengel
, “
Gutterballs and rolling down the tubes
,” Am. J. Phys.
89
, 459
–464
(2021
).13.
The ball is obviously neither homogeneous nor a perfect sphere; we would expect K to be different from 2/5 and 2/3. However, the experimental study gave a value close to 0.4 (cf. 4.1).
14.
Consider a mobile point M (position
, velocity
in the time derivative of the angular momentum; O is the origin of the lab frame, viz., a fixed point in this frame. Since
, where
is the total linear momentum, we have, when taking the time derivative of
,
. Moreover, the torque about M of all the forces
is
. Because
, we obtain for the angular momentum theorem about a mobile point
. This explains the term
in Eq. (8).
15.
16.
R. L.
Chaplin
and
M. G.
Miller
, “
Coefficient of friction for a sphere
,” Am. J. Phys.
52
, 1108
–1111
(1984
).17.
D. E.
Shaw
and
F. J.
Wunderlich
, “
Study of the slipping of a rolling sphere
,” Am. J. Phys.
52
, 997
–1000
(1984
).18.
A.
Doménech
,
T.
Doménech
, and
J.
Cebrián
, “
Introduction to the study of rolling friction
,” Am. J. Phys.
55
, 231
–235
(1987
).19.
R.
Cross
, “
Coulomb's law for rolling friction
,” Am. J. Phys.
84
, 221
–230
(2016
).20.
L.
Minkin
and
D.
Sikes
, “
Coefficient of rolling friction—Lab experiment
,” Am. J. Phys.
86
, 77
–78
(2018
).21.
R.
Feynman
,
R.
Leighton
, and
M.
Sands
, The Feynman Lectures on Physics
(
Addison-Wesley
,
Redwood City
, 2013
), Vol.
1
, Chap. 12.22.
P.
Appell
and
S.
Dautheville
, Précis de Mécanique Rationnelle
(
Gauthier-Villars
,
Paris
, 1934), p. 398
(in French).23.
See <https://en.wikipedia.org/wiki/Rolling_resistance> for information about the factor of rolling friction, in particular the Section Physical formulae, and the numerous references therein (in this webpage).
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2022
Author(s)
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